This distribution was first described by the German statistician Friedrich Robert Helmert in papers of 1875-6,[6][7] where he computed the sampling distribution of the sample variance of a normal population. Thus in German this was traditionally known as the Helmertsche ("Helmertian") or "Helmert distribution".

It follows from the definition of the chi-squared distribution that the sum of independent chi-squared variables is also chi-squared distributed. Specifically, if {Xi}i=1n are independent chi-squared variables with {ki}i=1n degrees of freedom, respectively, then Y = X1 + ⋯ + Xn is chi-squared distributed with k1 + ⋯ + kn degrees of freedom.

The sample mean of n i.i.d. chi-squared variables of degree k is distributed according to a gamma distribution with shape α and scale θ parameters:

Asymptotically, given that for a scale parameter going to infinity, a Gamma distribution converges towards a Normal distribution with expectation and variance , the sample mean converges towards:

Note that we would have obtained the same result invoking instead the central limit theorem, noting that for each chi-squared variable of degree the expectation is , and its variance (and hence the variance of the sample mean being ).

By the central limit theorem, because the chi-squared distribution is the sum of k independent random variables with finite mean and variance, it converges to a normal distribution for large k. For many practical purposes, for k > 50 the distribution is sufficiently close to a normal distribution for the difference to be ignored.[12] Specifically, if X ~ χ²(k), then as k tends to infinity, the distribution of tends to a standard normal distribution. However, convergence is slow as the skewness is and the excess kurtosis is 12/k.

The sampling distribution of ln(χ2) converges to normality much faster than the sampling distribution of χ2,[13] as the logarithm removes much of the asymmetry.[14] Other functions of the chi-squared distribution converge more rapidly to a normal distribution. Some examples are:

If X ~ χ²(k) then is approximately normally distributed with mean and unit variance (result credited to R. A. Fisher).

If X ~ χ²(k) then is approximately normally distributed with mean and variance [15] This is known as the Wilson–Hilferty transformation.

The chi-squared distribution is obtained as the sum of the squares of k independent, zero-mean, unit-variance Gaussian random variables. Generalizations of this distribution can be obtained by summing the squares of other types of Gaussian random variables. Several such distributions are described below.

If are chi square random variables and , then a closed expression for the distribution of is not known. It may be, however, calculated using the property of characteristic functions of the chi-squared random variable.[16]

The chi-squared distribution X ~ χ²(k) is a special case of the gamma distribution, in that X ~ Γ(k/2, 1/2) using the rate parameterization of the gamma distribution (or X ~ Γ(k/2, 2) using the scale parameterization of the gamma distribution) where k is an integer.

The Erlang distribution is also a special case of the Gamma distribution and thus we also have that if X ~ χ²(k) with even k, then X is Erlang distributed with shape parameter k/2 and scale parameter 1/2.

The p-value is the probability of observing a test statistic at least as extreme in a chi-squared distribution. Accordingly, since the cumulative distribution function (CDF) for the appropriate degrees of freedom (df) gives the probability of having obtained a value less extreme than this point, subtracting the CDF value from 1 gives the p-value. The table below gives a number of p-values matching to χ2 for the first 10 degrees of freedom.

A low p-value indicates greater statistical significance, i.e. greater confidence that the observed deviation from the null hypothesis is significant. A p-value of 0.05 is often used as a bright-line cutoff between significant and not-significant results.

^Bartlett, M. S.; Kendall, D. G. (1946). "The Statistical Analysis of Variance-Heterogeneity and the Logarithmic Transformation". Supplement to the Journal of the Royal Statistical Society8 (1): 128–138. JSTOR2983618.